User talk:Egm6321.f10.team5.steinberg/hw7

Part 1
The Legendre polynomials $$ P_n \! $$ are even when n is even and odd when n is odd (see Problem 7.3 above). That is to say that

$$ P_{2k} \! $$ is EVEN

and

$$ P_{2k+1} \! $$ is ODD for $$ k = 0, 1, 2, 3 ... $$

Mathematically, the product of 2 odd functions is an even function, the product of 2 even functions is an even function and the product of an even function with an odd function is an odd function.

If the integrand of $$ A_n $$ is odd, the integral will be zero since our limits of integration equally straddle $$ \mu = 0 $$. However, if the integrand is even, $$ A_n $$ will be non-zero. We have already examined the even and odd properties of the Legendre polynomials $$ P_n $$. Thus, we must now look at $$ f(\mu) $$ to determine the even and odd properties of $$ A_n $$.

Part 1a
where we recall that $$ \mu := sin(\theta) \! $$

Equation (4.3) is a 6th order polynomial in $$ \mu $$, therefore it is an even function.

Therefore, with this $$ f(\mu) $$ we may conclude the following:

The reason k stops at 3 is that for $$ n \ge 7 $$, $$ A_n $$ is zero, since all Legendre polynomials at or beyond that order are orthogonal to $$ f $$.

Part 1b
For this part, we will examine $$ A_n $$ as a function of $$ \theta $$ since we cannot easily make f an explicit function of $$ \mu $$.

By merely looking at the behavior of $$ f(\theta) $$ at the limits of integration and at zero, it is apparent that it is an even function.

Therefore, we observe that

Part 2a
We now evaluate the non-zero values of $$ A_n $$ using Wolfram Alpha.

Part 2b
We now evaluate the few non-zero values of $$ A_n $$ using Wolfram Alpha.